I did teach both my children to count on their fingers using the pattern taught by ASL. I've noticed that sometimes her friends look at her oddly when she holds up her index, middle, and thumb for the number three, and obviously, they have no clue that holding up the index, middle, and ring fingers might mean either six or three to her depending on to whom she is talking.
I introduced both my children to an awesome game when they were just in kindergarten: http://www.dragonboxapp.com/ it teaches basic algebraic concepts cloaked in the guise of a fun puzzle game.
I was first taught to count starting from the index finger as well:
fist = 0
index = 1
index, middle = 2
index, middle, ring = 3
index, middle, ring, pinky = 4
index, middle, ring, pinky, thumb = 5
thumb = 6
thumb, index = 7
thumb, index, middle = 8
thumb, index, middle, ring = 9
(I am from Thailand, but I think this is how counting is taught in America as well?)
I thought this was universal until I went to Singapore. I was quite intrigued that in Singapore a "thumb" is not considered a number but rather a signal for "good" and instead they used thumb + pinky for 6. (The rest seems to be the same.)
Not sure why I brought that up, but I guess I had not realized that there was a difference in even the way we counted all around the world until that point.
EDIT: Formatting
There are a few other fancy ways including even binary which could let you count up to 1024 if you could keep it all in your head. :)
The ASL (American Sign Language) method I mentioned in the parent is described here: http://www.dummies.com/how-to/content/counting-on-numbers-in...
I was a precocious little shit.
fist = 0 index = 1 index, middle = 2 index, middle, ring = 3 index, middle, ring, pinky = 4 index, middle, ring, pinky, thumb splayed = 5 thumb, pinky = 6 index, middle, ring, pinky, thumb all held strait with their pads touching, looking a bit like a shadow puppet of a bird's head = 7 thumb, index = 8 just the index finger, bent in half = 9
Apart from 4 and 5, you can kind of see the link between the written form of the number and the hand signal: 1, 2, 3: obvious because 一二三 6: the thumb and pinky make up the bottom part of the character 六 7: If you look at your own right hand upside-down when making the gesture, the bird's head points in the same direction as the 七 8: Your thumb and index finger are the two lines in 八 9: Your index finger is the second stroke in 九
I don't know whether this is coincidental, or even just my imagination...
Then, when we started playing board games there was an issue playing games like "Game of Goose" because she started counting with 1 in the occupied space. We "fixed" that starting with 0.
Hmm interesting. I have been trying something different as well. I have been counting with fingers with my kid using different fingers every time. The idea is that it doesn't matter to associate a finger with a number as long as the correct number of fingers is up. Like say most people start with the thumb and move left or right. Sometimes I start with the ring finger.
Counting is also fun in general. We count steps, count trees when we drive. I try to never make it about teaching but more about fun.
I'm intrigued by your 0 vs. 1 experiment. I wonder if there is something fundamentally different in how we learn the concept of ordinal numbers (1, 2, 3) versus, say, integers (-1, 0, 1). Usually we teach numbers by counting physical objects, and perhaps it takes a little longer to understand the concept of zero if you're used to counting objects.